Wow, thanks everyone for giving so many and such varied answers! This is excellent help and advice!
I'd like to give you the good news: I found the solution (at least the cause and a mostly complete solution)! Technically, my doctor figured it out.
For me, the solution was to take prescription wakefulness medication (like they give people with narcolepsy or sleep apnea that can't be controlled any other way). This is because my exhaustion was the side-effect of some other medicines that I have to take. No amount of other intervention (non-medication) will be able to fix this problem, the doc said, although I'm going to try a bunch of the things you all suggested because I'm still pretty tired and unmotivated during the day and I think several suggestions will help with that remaining tiredness without having to increase my medicine dosage.
Thanks again for answering my question, and I'm glad to see that several hubs have been written about this, too. (I may write one myself.) I'm going to look into most of the solutions you've recommended.
I'll keep you posted on which of the solutions you all recommended work the best for me! (Meditation... I have a "singing bowl" and used to meditate to gain energy, iron deficiency could be it, too...)